LMFDB - Embedded newform 869.1.j.a.394.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [869,1,Mod(157,869)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(869, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([8, 5]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("869.157");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 869 = 11 \cdot 79 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	869.j (of order $10$, degree $4$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.433687495978$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$5$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{50})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{15} + x^{10} - x^{5} + 1 $ x^20 - x^15 + x^10 - x^5 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{25}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{25} - \cdots)$

Embedding invariants

Embedding label			394.1
Root			$-0.876307 + 0.481754i$ of defining polynomial
Character	$\chi$	$=$	869.394
Dual form			869.1.j.a.236.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.613161 + 1.88711i) q^{2} +(-2.37622 - 1.72642i) q^{4} +(0.0388067 + 0.119435i) q^{5} +(3.10969 - 2.25932i) q^{8} +(0.309017 - 0.951057i) q^{9} +O(q^{10})$	\(q+(-0.613161 + 1.88711i) q^{2} +(-2.37622 - 1.72642i) q^{4} +(0.0388067 + 0.119435i) q^{5} +(3.10969 - 2.25932i) q^{8} +(0.309017 - 0.951057i) q^{9} -0.249182 q^{10} +(-0.992115 - 0.125333i) q^{11} +(0.541587 - 1.66683i) q^{13} +(1.44922 + 4.46025i) q^{16} +(1.60528 + 1.16630i) q^{18} +(1.03137 - 0.749337i) q^{19} +(0.113982 - 0.350799i) q^{20} +(0.844844 - 1.79538i) q^{22} -1.27485 q^{23} +(0.796258 - 0.578516i) q^{25} +(2.81343 + 2.04407i) q^{26} +(-0.263146 + 0.809880i) q^{31} -5.46182 q^{32} +(-2.37622 + 1.72642i) q^{36} +(0.781687 + 2.40578i) q^{38} +(0.390518 + 0.283728i) q^{40} +(2.14110 + 2.01063i) q^{44} +0.125581 q^{45} +(0.781687 - 2.40578i) q^{46} +(0.309017 + 0.951057i) q^{49} +(0.603491 + 1.85735i) q^{50} +(-4.16459 + 3.02575i) q^{52} +(-0.0235315 - 0.123357i) q^{55} +(-1.36699 - 0.993173i) q^{62} +(1.89975 - 5.84683i) q^{64} +0.220095 q^{65} +1.07165 q^{67} +(-1.18779 - 3.65565i) q^{72} +(-1.56720 - 1.13864i) q^{73} -3.74444 q^{76} +(0.309017 - 0.951057i) q^{79} +(-0.476468 + 0.346175i) q^{80} +(-0.809017 - 0.587785i) q^{81} +(0.190983 + 0.587785i) q^{83} +(-3.36833 + 1.85176i) q^{88} +1.45794 q^{89} +(-0.0770013 + 0.236986i) q^{90} +(3.02932 + 2.20093i) q^{92} +(0.129521 + 0.0941025i) q^{95} +(-0.574633 + 1.76854i) q^{97} -1.98423 q^{98} +(-0.425779 + 0.904827i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 5 q^{4} - 5 q^{9}+O(q^{10}) $ 20 * q - 5 * q^4 - 5 * q^9	$ 20 q - 5 q^{4} - 5 q^{9} - 5 q^{16} + 15 q^{20} - 5 q^{22} - 5 q^{25} + 15 q^{26} - 10 q^{32} - 5 q^{36} - 10 q^{40} - 5 q^{49} - 10 q^{50} - 10 q^{62} - 5 q^{64} - 10 q^{76} - 5 q^{79} - 10 q^{80} - 5 q^{81} + 15 q^{83} - 5 q^{88} + 15 q^{92} + 15 q^{95}+O(q^{100}) $ 20 * q - 5 * q^4 - 5 * q^9 - 5 * q^16 + 15 * q^20 - 5 * q^22 - 5 * q^25 + 15 * q^26 - 10 * q^32 - 5 * q^36 - 10 * q^40 - 5 * q^49 - 10 * q^50 - 10 * q^62 - 5 * q^64 - 10 * q^76 - 5 * q^79 - 10 * q^80 - 5 * q^81 + 15 * q^83 - 5 * q^88 + 15 * q^92 + 15 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/869\mathbb{Z}\right)^\times$.

$n$	$475$	$793$
$\chi(n)$	$e\left(\frac{3}{5}\right)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.187381	+	0.982287i	$0.560000\pi$
$2$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.425779	+	0.904827i	$0.640000\pi$
$3$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$3$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$4$	−2.37622	−	1.72642i	−2.37622	−	1.72642i
$5$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	0.968583	−	0.248690i	$-0.0800000\pi$
$5$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	−0.929776	+	0.368125i	$0.880000\pi$
$6$		0			0
$7$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$7$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$8$	3.10969	−	2.25932i	3.10969	−	2.25932i
$9$	0.309017	−	0.951057i	0.309017	−	0.951057i
$10$	−0.249182			−0.249182
$11$	−0.992115	−	0.125333i	−0.992115	−	0.125333i
$11$	−0.992115	−	0.125333i	−0.992115	−	0.125333i
$12$		0			0
$13$	0.541587	−	1.66683i	0.541587	−	1.66683i	−0.187381	−	0.982287i	$-0.560000\pi$
$13$	0.541587	−	1.66683i	0.541587	−	1.66683i	0.728969	−	0.684547i	$-0.240000\pi$
$14$		0			0
$15$		0			0
$16$	1.44922	+	4.46025i	1.44922	+	4.46025i
$17$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$17$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$18$	1.60528	+	1.16630i	1.60528	+	1.16630i
$19$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.0627905	−	0.998027i	$-0.480000\pi$
$19$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.968583	+	0.248690i	$0.0800000\pi$
$20$	0.113982	−	0.350799i	0.113982	−	0.350799i
$21$		0			0
$22$	0.844844	−	1.79538i	0.844844	−	1.79538i
$23$	−1.27485			−1.27485			−0.637424	−	0.770513i	$-0.720000\pi$
$23$	−1.27485			−1.27485			−0.637424	+	0.770513i	$0.720000\pi$
$24$		0			0
$25$	0.796258	−	0.578516i	0.796258	−	0.578516i
$26$	2.81343	+	2.04407i	2.81343	+	2.04407i
$27$		0			0
$28$		0			0
$29$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$29$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$30$		0			0
$31$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	0.728969	+	0.684547i	$0.240000\pi$
$31$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	−0.992115	+	0.125333i	$0.960000\pi$
$32$	−5.46182			−5.46182
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−2.37622	+	1.72642i	−2.37622	+	1.72642i
$37$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$37$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$38$	0.781687	+	2.40578i	0.781687	+	2.40578i
$39$		0			0
$40$	0.390518	+	0.283728i	0.390518	+	0.283728i
$41$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$41$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	2.14110	+	2.01063i	2.14110	+	2.01063i
$45$	0.125581			0.125581
$46$	0.781687	−	2.40578i	0.781687	−	2.40578i
$47$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$47$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$48$		0			0
$49$	0.309017	+	0.951057i	0.309017	+	0.951057i
$50$	0.603491	+	1.85735i	0.603491	+	1.85735i
$51$		0			0
$52$	−4.16459	+	3.02575i	−4.16459	+	3.02575i
$53$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$53$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$54$		0			0
$55$	−0.0235315	−	0.123357i	−0.0235315	−	0.123357i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$59$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$60$		0			0
$61$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$61$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$62$	−1.36699	−	0.993173i	−1.36699	−	0.993173i
$63$		0			0
$64$	1.89975	−	5.84683i	1.89975	−	5.84683i
$65$	0.220095			0.220095
$66$		0			0
$67$	1.07165			1.07165			0.535827	−	0.844328i	$-0.320000\pi$
$67$	1.07165			1.07165			0.535827	+	0.844328i	$0.320000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$71$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$72$	−1.18779	−	3.65565i	−1.18779	−	3.65565i
$73$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.929776	−	0.368125i	$-0.880000\pi$
$73$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.637424	−	0.770513i	$-0.720000\pi$
$74$		0			0
$75$		0			0
$76$	−3.74444			−3.74444
$77$		0			0
$78$		0			0
$79$	0.309017	−	0.951057i	0.309017	−	0.951057i
$79$	0.309017	−	0.951057i	0.309017	−	0.951057i
$80$	−0.476468	+	0.346175i	−0.476468	+	0.346175i
$81$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$82$		0			0
$83$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$83$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−3.36833	+	1.85176i	−3.36833	+	1.85176i
$89$	1.45794			1.45794			0.728969	−	0.684547i	$-0.240000\pi$
$89$	1.45794			1.45794			0.728969	+	0.684547i	$0.240000\pi$
$90$	−0.0770013	+	0.236986i	−0.0770013	+	0.236986i
$91$		0			0
$92$	3.02932	+	2.20093i	3.02932	+	2.20093i
$93$		0			0
$94$		0			0
$95$	0.129521	+	0.0941025i	0.129521	+	0.0941025i
$96$		0			0
$97$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	0.0627905	+	0.998027i	$0.480000\pi$
$97$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	−0.637424	+	0.770513i	$0.720000\pi$
$98$	−1.98423			−1.98423
$99$	−0.425779	+	0.904827i	−0.425779	+	0.904827i
$100$	−2.89085			−2.89085
$101$	0.450527	−	1.38658i	0.450527	−	1.38658i	−0.425779	−	0.904827i	$-0.640000\pi$
$101$	0.450527	−	1.38658i	0.450527	−	1.38658i	0.876307	−	0.481754i	$-0.160000\pi$
$102$		0			0
$103$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$103$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$104$	−2.08174	−	6.40695i	−2.08174	−	6.40695i
$105$		0			0
$106$		0			0
$107$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$107$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	0.247217	+	0.0312307i	0.247217	+	0.0312307i
$111$		0			0
$112$		0			0
$113$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$113$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$114$		0			0
$115$	−0.0494726	−	0.152261i	−0.0494726	−	0.152261i
$116$		0			0
$117$	−1.41789	−	1.03016i	−1.41789	−	1.03016i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.968583	+	0.248690i	0.968583	+	0.248690i
$122$		0			0
$123$		0			0
$124$	2.02349	−	1.47015i	2.02349	−	1.47015i
$125$	0.201592	+	0.146465i	0.201592	+	0.146465i
$126$		0			0
$127$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$127$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$128$	5.45008	+	3.95971i	5.45008	+	3.95971i
$129$		0			0
$130$	−0.134954	+	0.415344i	−0.134954	+	0.415344i
$131$	1.93717			1.93717			0.968583	−	0.248690i	$-0.0800000\pi$
$131$	1.93717			1.93717			0.968583	+	0.248690i	$0.0800000\pi$
$132$		0			0
$133$		0			0
$134$	−0.657096	+	2.02233i	−0.657096	+	2.02233i
$135$		0			0
$136$		0			0
$137$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$137$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$138$		0			0
$139$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$139$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.746226	+	1.58581i	−0.746226	+	1.58581i
$144$	4.68978			4.68978
$145$		0			0
$146$	3.10969	−	2.25932i	3.10969	−	2.25932i
$147$		0			0
$148$		0			0
$149$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$149$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$150$		0			0
$151$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	−0.929776	−	0.368125i	$-0.880000\pi$
$151$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	0.0627905	+	0.998027i	$0.480000\pi$
$152$	1.51426	−	4.66040i	1.51426	−	4.66040i
$153$		0			0
$154$		0			0
$155$	−0.106940			−0.106940
$156$		0			0
$157$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$157$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$158$	1.60528	+	1.16630i	1.60528	+	1.16630i
$159$		0			0
$160$	−0.211955	−	0.652330i	−0.211955	−	0.652330i
$161$		0			0
$162$	1.60528	−	1.16630i	1.60528	−	1.16630i
$163$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	−0.992115	−	0.125333i	$-0.960000\pi$
$163$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	0.876307	+	0.481754i	$0.160000\pi$
$164$		0			0
$165$		0			0
$166$	−1.22632			−1.22632
$167$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	0.728969	+	0.684547i	$0.240000\pi$
$167$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	−0.992115	+	0.125333i	$0.960000\pi$
$168$		0			0
$169$	−1.67600	−	1.21769i	−1.67600	−	1.21769i
$170$		0			0
$171$	−0.393950	−	1.21245i	−0.393950	−	1.21245i
$172$		0			0
$173$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$173$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$174$		0			0
$175$		0			0
$176$	−0.878777	−	4.60671i	−0.878777	−	4.60671i
$177$		0			0
$178$	−0.893950	+	2.75129i	−0.893950	+	2.75129i
$179$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$179$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$180$	−0.298408	−	0.216806i	−0.298408	−	0.216806i
$181$	0.541587	+	1.66683i	0.541587	+	1.66683i	0.728969	+	0.684547i	$0.240000\pi$
$181$	0.541587	+	1.66683i	0.541587	+	1.66683i	−0.187381	+	0.982287i	$0.560000\pi$
$182$		0			0
$183$		0			0
$184$	−3.96438	+	2.88029i	−3.96438	+	2.88029i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	−0.256999	+	0.186721i	−0.256999	+	0.186721i
$191$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$191$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$192$		0			0
$193$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$193$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$194$	−2.98509	−	2.16880i	−2.98509	−	2.16880i
$195$		0			0
$196$	0.907634	−	2.79341i	0.907634	−	2.79341i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−1.44644	−	1.35830i	−1.44644	−	1.35830i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	1.16906	−	3.59800i	1.16906	−	3.59800i
$201$		0			0
$202$	2.34039	+	1.70039i	2.34039	+	1.70039i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.393950	+	1.21245i	−0.393950	+	1.21245i
$208$	8.21937			8.21937
$209$	−1.11716	+	0.614163i	−1.11716	+	0.614163i
$210$		0			0
$211$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$211$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.157050	+	0.333748i	−0.157050	+	0.333748i
$221$		0			0
$222$		0			0
$223$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$223$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$224$		0			0
$225$	−0.304144	−	0.936058i	−0.304144	−	0.936058i
$226$		0			0
$227$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$227$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$228$		0			0
$229$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$229$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$230$	0.317669			0.317669
$231$		0			0
$232$		0			0
$233$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$233$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$234$	2.81343	−	2.04407i	2.81343	−	2.04407i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	−0.637424	−	0.770513i	$-0.720000\pi$
$239$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	0.535827	+	0.844328i	$0.320000\pi$
$240$		0			0
$241$	−0.374763			−0.374763			−0.187381	−	0.982287i	$-0.560000\pi$
$241$	−0.374763			−0.374763			−0.187381	+	0.982287i	$0.560000\pi$
$242$	−1.06320	+	1.67534i	−1.06320	+	1.67534i
$243$		0			0
$244$		0			0
$245$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i
$246$		0			0
$247$	−0.690441	−	2.12496i	−0.690441	−	2.12496i
$248$	1.01148	+	3.11300i	1.01148	+	3.11300i
$249$		0			0
$250$	−0.400005	+	0.290621i	−0.400005	+	0.290621i
$251$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$251$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$252$		0			0
$253$	1.26480	+	0.159781i	1.26480	+	0.159781i
$254$		0			0
$255$		0			0
$256$	−5.84059	+	4.24344i	−5.84059	+	4.24344i
$257$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.992115	−	0.125333i	$-0.960000\pi$
$257$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.425779	−	0.904827i	$-0.640000\pi$
$258$		0			0
$259$		0			0
$260$	−0.522994	−	0.379977i	−0.522994	−	0.379977i
$261$		0			0
$262$	−1.18779	+	3.65565i	−1.18779	+	3.65565i
$263$	−0.374763			−0.374763			−0.187381	−	0.982287i	$-0.560000\pi$
$263$	−0.374763			−0.374763			−0.187381	+	0.982287i	$0.560000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−2.54648	−	1.85013i	−2.54648	−	1.85013i
$269$	0.331159	+	1.01920i	0.331159	+	1.01920i	0.968583	+	0.248690i	$0.0800000\pi$
$269$	0.331159	+	1.01920i	0.331159	+	1.01920i	−0.637424	+	0.770513i	$0.720000\pi$
$270$		0			0
$271$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$271$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.862487	+	0.474156i	−0.862487	+	0.474156i
$276$		0			0
$277$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	−0.929776	−	0.368125i	$-0.880000\pi$
$277$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	0.968583	+	0.248690i	$0.0800000\pi$
$278$		0			0
$279$	0.688925	+	0.500534i	0.688925	+	0.500534i
$280$		0			0
$281$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	0.968583	−	0.248690i	$-0.0800000\pi$
$281$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	−0.929776	+	0.368125i	$0.880000\pi$
$282$		0			0
$283$	0.688925	−	0.500534i	0.688925	−	0.500534i	−0.187381	−	0.982287i	$-0.560000\pi$
$283$	0.688925	−	0.500534i	0.688925	−	0.500534i	0.876307	+	0.481754i	$0.160000\pi$
$284$		0			0
$285$		0			0
$286$	−2.53505	−	2.38057i	−2.53505	−	2.38057i
$287$		0			0
$288$	−1.68779	+	5.19450i	−1.68779	+	5.19450i
$289$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$290$		0			0
$291$		0			0
$292$	1.75824	+	5.41130i	1.75824	+	5.41130i
$293$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$293$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.690441	+	2.12496i	−0.690441	+	2.12496i
$300$		0			0
$301$		0			0
$302$	−0.657096	−	2.02233i	−0.657096	−	2.02233i
$303$		0			0
$304$	4.83692	+	3.51422i	4.83692	+	3.51422i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$	0.0655712	−	0.201807i	0.0655712	−	0.201807i
$311$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$311$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$312$		0			0
$313$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	−0.992115	−	0.125333i	$-0.960000\pi$
$313$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	0.728969	−	0.684547i	$-0.240000\pi$
$314$		0			0
$315$		0			0
$316$	−2.37622	+	1.72642i	−2.37622	+	1.72642i
$317$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$317$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$318$		0			0
$319$		0			0
$320$	0.772037			0.772037
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.907634	+	2.79341i	0.907634	+	2.79341i
$325$	−0.533046	−	1.64055i	−0.533046	−	1.64055i
$326$	−0.601597	−	0.437086i	−0.601597	−	0.437086i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	0.560949	−	1.72642i	0.560949	−	1.72642i
$333$		0			0
$334$	−1.36699	−	0.993173i	−1.36699	−	0.993173i
$335$	0.0415873	+	0.127993i	0.0415873	+	0.127993i
$336$		0			0
$337$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.968583	+	0.248690i	$0.0800000\pi$
$337$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.535827	+	0.844328i	$0.320000\pi$
$338$	3.32557	−	2.41617i	3.32557	−	2.41617i
$339$		0			0
$340$		0			0
$341$	0.362576	−	0.770513i	0.362576	−	0.770513i
$342$	2.52959			2.52959
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.331159	+	1.01920i	0.331159	+	1.01920i	0.968583	+	0.248690i	$0.0800000\pi$
$347$	0.331159	+	1.01920i	0.331159	+	1.01920i	−0.637424	+	0.770513i	$0.720000\pi$
$348$		0			0
$349$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$349$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$350$		0			0
$351$		0			0
$352$	5.41875	+	0.684547i	5.41875	+	0.684547i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$	−3.46438	−	2.51702i	−3.46438	−	2.51702i
$357$		0			0
$358$	−0.378954	−	1.16630i	−0.378954	−	1.16630i
$359$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$359$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$360$	0.390518	−	0.283728i	0.390518	−	0.283728i
$361$	0.193209	−	0.594636i	0.193209	−	0.594636i
$362$	−3.47759			−3.47759
$363$		0			0
$364$		0			0
$365$	0.0751750	−	0.231365i	0.0751750	−	0.231365i
$366$		0			0
$367$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.187381	−	0.982287i	$-0.560000\pi$
$367$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.992115	+	0.125333i	$0.960000\pi$
$368$	−1.84754	−	5.68613i	−1.84754	−	5.68613i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$379$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$380$	−0.145309	−	0.447216i	−0.145309	−	0.447216i
$381$		0			0
$382$		0			0
$383$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.187381	+	0.982287i	$0.560000\pi$
$383$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.425779	+	0.904827i	$0.640000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	4.41870	−	3.21038i	4.41870	−	3.21038i
$389$	0.688925	+	0.500534i	0.688925	+	0.500534i	0.876307	−	0.481754i	$-0.160000\pi$
$389$	0.688925	+	0.500534i	0.688925	+	0.500534i	−0.187381	+	0.982287i	$0.560000\pi$
$390$		0			0
$391$		0			0
$392$	3.10969	+	2.25932i	3.10969	+	2.25932i
$393$		0			0
$394$		0			0
$395$	0.125581			0.125581
$396$	2.57386	−	1.41499i	2.57386	−	1.41499i
$397$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$397$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$398$		0			0
$399$		0			0
$400$	3.73428	+	2.71311i	3.73428	+	2.71311i
$401$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$401$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$402$		0			0
$403$	1.20742	+	0.877242i	1.20742	+	0.877242i
$404$	−3.46438	+	2.51702i	−3.46438	+	2.51702i
$405$	0.0388067	−	0.119435i	0.0388067	−	0.119435i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$409$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−2.04648	−	1.48686i	−2.04648	−	1.48686i
$415$	−0.0627905	+	0.0456200i	−0.0627905	+	0.0456200i
$416$	−2.95805	+	9.10394i	−2.95805	+	9.10394i
$417$		0			0
$418$	−0.473998	−	2.48478i	−0.473998	−	2.48478i
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.535827	−	0.844328i	$-0.320000\pi$
$421$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.968583	−	0.248690i	$-0.0800000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.0627905	−	0.998027i	$-0.480000\pi$
$431$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.535827	−	0.844328i	$-0.320000\pi$
$432$		0			0
$433$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.968583	−	0.248690i	$-0.0800000\pi$
$433$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.0627905	+	0.998027i	$0.480000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.31484	+	0.955291i	−1.31484	+	0.955291i
$438$		0			0
$439$	−1.85955			−1.85955			−0.929776	−	0.368125i	$-0.880000\pi$
$439$	−1.85955			−1.85955			−0.929776	+	0.368125i	$0.880000\pi$
$440$	−0.351878	−	0.330435i	−0.351878	−	0.330435i
$441$	1.00000			1.00000
$442$		0			0
$443$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$443$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$444$		0			0
$445$	0.0565777	+	0.174128i	0.0565777	+	0.174128i
$446$	−0.378954	−	1.16630i	−0.378954	−	1.16630i
$447$		0			0
$448$		0			0
$449$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$449$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$450$	1.95294			1.95294
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.450527	+	1.38658i	0.450527	+	1.38658i	0.876307	+	0.481754i	$0.160000\pi$
$457$	0.450527	+	1.38658i	0.450527	+	1.38658i	−0.425779	+	0.904827i	$0.640000\pi$
$458$		0			0
$459$		0			0
$460$	−0.145309	+	0.447216i	−0.145309	+	0.447216i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.425779	−	0.904827i	$-0.640000\pi$
$467$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.187381	−	0.982287i	$-0.560000\pi$
$468$	1.59073	+	4.89577i	1.59073	+	4.89577i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.387737	−	1.19333i	0.387737	−	1.19333i
$476$		0			0
$477$		0			0
$478$	−0.0770013	−	0.236986i	−0.0770013	−	0.236986i
$479$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$479$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$480$		0			0
$481$		0			0
$482$	0.229790	−	0.707220i	0.229790	−	0.707220i
$483$		0			0
$484$	−1.87222	−	2.26313i	−1.87222	−	2.26313i
$485$	−0.233525			−0.233525
$486$		0			0
$487$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.425779	+	0.904827i	$0.640000\pi$
$487$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.992115	+	0.125333i	$0.960000\pi$
$488$		0			0
$489$		0			0
$490$	−0.0770013	−	0.236986i	−0.0770013	−	0.236986i
$491$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$491$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$492$		0			0
$493$		0			0
$494$	4.43339			4.43339
$495$	−0.124591	−	0.0157395i	−0.124591	−	0.0157395i
$496$	−3.99362			−3.99362
$497$		0			0
$498$		0			0
$499$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.876307	+	0.481754i	$0.160000\pi$
$499$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.728969	+	0.684547i	$0.240000\pi$
$500$	−0.226166	−	0.696067i	−0.226166	−	0.696067i
$501$		0			0
$502$		0			0
$503$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$503$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$504$		0			0
$505$	0.183089			0.183089
$506$	−1.07705	+	2.28884i	−1.07705	+	2.28884i
$507$		0			0
$508$		0			0
$509$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$509$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$510$		0			0
$511$		0			0
$512$	−2.34489	−	7.21683i	−2.34489	−	7.21683i
$513$		0			0
$514$	2.81343	−	2.04407i	2.81343	−	2.04407i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	0.684426	−	0.497265i	0.684426	−	0.497265i
$521$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$521$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$522$		0			0
$523$	0.450527	+	1.38658i	0.450527	+	1.38658i	0.876307	+	0.481754i	$0.160000\pi$
$523$	0.450527	+	1.38658i	0.450527	+	1.38658i	−0.425779	+	0.904827i	$0.640000\pi$
$524$	−4.60313	−	3.34437i	−4.60313	−	3.34437i
$525$		0			0
$526$	0.229790	−	0.707220i	0.229790	−	0.707220i
$527$		0			0
$528$		0			0
$529$	0.625237			0.625237
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	3.33251	−	2.42121i	3.33251	−	2.42121i
$537$		0			0
$538$	−2.12641			−2.12641
$539$	−0.187381	−	0.982287i	−0.187381	−	0.982287i
$540$		0			0
$541$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$541$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.303189	−	0.220280i	0.303189	−	0.220280i	−0.425779	−	0.904827i	$-0.640000\pi$
$547$	0.303189	−	0.220280i	0.303189	−	0.220280i	0.728969	+	0.684547i	$0.240000\pi$
$548$		0			0
$549$		0			0
$550$	−0.365944	−	1.91835i	−0.365944	−	1.91835i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.201592	+	0.146465i	0.201592	+	0.146465i
$555$		0			0
$556$		0			0
$557$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.968583	−	0.248690i	$-0.0800000\pi$
$557$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.0627905	+	0.998027i	$0.480000\pi$
$558$	−1.36699	+	0.993173i	−1.36699	+	0.993173i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.249182			−0.249182
$563$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$563$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$564$		0			0
$565$		0			0
$566$	0.522142	+	1.60699i	0.522142	+	1.60699i
$567$		0			0
$568$		0			0
$569$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.535827	−	0.844328i	$-0.320000\pi$
$569$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.968583	−	0.248690i	$-0.0800000\pi$
$570$		0			0
$571$	−0.851559			−0.851559			−0.425779	−	0.904827i	$-0.640000\pi$
$571$	−0.851559			−0.851559			−0.425779	+	0.904827i	$0.640000\pi$
$572$	4.51098	−	2.47993i	4.51098	−	2.47993i
$573$		0			0
$574$		0			0
$575$	−1.01511	+	0.737519i	−1.01511	+	0.737519i
$576$	−4.97361	−	3.61354i	−4.97361	−	3.61354i
$577$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$577$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$578$	−0.613161	−	1.88711i	−0.613161	−	1.88711i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−7.44605			−7.44605
$585$	0.0680131	−	0.209323i	0.0680131	−	0.209323i
$586$		0			0
$587$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$587$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$588$		0			0
$589$	0.335471	+	1.03247i	0.335471	+	1.03247i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.125581			0.125581			0.0627905	−	0.998027i	$-0.480000\pi$
$593$	0.125581			0.125581			0.0627905	+	0.998027i	$0.480000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−3.58669	−	2.60588i	−3.58669	−	2.60588i
$599$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.535827	+	0.844328i	$0.320000\pi$
$599$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.0627905	+	0.998027i	$0.480000\pi$
$600$		0			0
$601$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$601$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$602$		0			0
$603$	0.331159	−	1.01920i	0.331159	−	1.01920i
$604$	3.14762			3.14762
$605$	0.00788530	+	0.125333i	0.00788530	+	0.125333i
$606$		0			0
$607$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$607$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$608$	−5.63317	+	4.09274i	−5.63317	+	4.09274i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$613$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.75261			1.75261			0.876307	−	0.481754i	$-0.160000\pi$
$617$	1.75261			1.75261			0.876307	+	0.481754i	$0.160000\pi$
$618$		0			0
$619$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$619$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$620$	0.254112	+	0.184623i	0.254112	+	0.184623i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.294474	−	0.906297i	0.294474	−	0.906297i
$626$	1.68969			1.68969
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$631$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$632$	−1.18779	−	3.65565i	−1.18779	−	3.65565i
$633$		0			0
$634$	−2.59739	−	1.88711i	−2.59739	−	1.88711i
$635$		0			0
$636$		0			0
$637$	1.75261			1.75261
$638$		0			0
$639$		0			0
$640$	−0.261428	+	0.804591i	−0.261428	+	0.804591i
$641$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.0627905	−	0.998027i	$-0.480000\pi$
$641$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.968583	+	0.248690i	$0.0800000\pi$
$642$		0			0
$643$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.535827	+	0.844328i	$0.320000\pi$
$643$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.0627905	+	0.998027i	$0.480000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$647$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$648$	−3.84378			−3.84378
$649$		0			0
$650$	3.42274			3.42274
$651$		0			0
$652$	0.890518	−	0.646999i	0.890518	−	0.646999i
$653$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.187381	−	0.982287i	$-0.560000\pi$
$653$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.992115	+	0.125333i	$0.960000\pi$
$654$		0			0
$655$	0.0751750	+	0.231365i	0.0751750	+	0.231365i
$656$		0			0
$657$	−1.56720	+	1.13864i	−1.56720	+	1.13864i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$	1.92189	+	1.39634i	1.92189	+	1.39634i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	2.02349	−	1.47015i	2.02349	−	1.47015i
$669$		0			0
$670$	−0.267036			−0.267036
$671$		0			0
$672$		0			0
$673$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$673$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$674$	−2.98509	+	2.16880i	−2.98509	+	2.16880i
$675$		0			0
$676$	1.88030	+	5.78698i	1.88030	+	5.78698i
$677$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$677$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	1.23173	+	1.15667i	1.23173	+	1.15667i
$683$	1.93717			1.93717			0.968583	−	0.248690i	$-0.0800000\pi$
$683$	1.93717			1.93717			0.968583	+	0.248690i	$0.0800000\pi$
$684$	−1.15710	+	3.56117i	−1.15710	+	3.56117i
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$691$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$692$		0			0
$693$		0			0
$694$	−2.12641			−2.12641
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$701$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$702$		0			0
$703$		0			0
$704$	−2.61757	+	5.56262i	−2.61757	+	5.56262i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$709$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$710$		0			0
$711$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$712$	4.53373	−	3.29394i	4.53373	−	3.29394i
$713$	0.335471	−	1.03247i	0.335471	−	1.03247i
$714$		0			0
$715$	−0.218359	−	0.0275852i	−0.218359	−	0.0275852i
$716$	1.81527			1.81527
$717$		0			0
$718$		0			0
$719$	0.303189	+	0.220280i	0.303189	+	0.220280i	0.728969	−	0.684547i	$-0.240000\pi$
$719$	0.303189	+	0.220280i	0.303189	+	0.220280i	−0.425779	+	0.904827i	$0.640000\pi$
$720$	0.181995	+	0.560122i	0.181995	+	0.560122i
$721$		0			0
$722$	1.00368	+	0.729215i	1.00368	+	0.729215i
$723$		0			0
$724$	1.59073	−	4.89577i	1.59073	−	4.89577i
$725$		0			0
$726$		0			0
$727$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$727$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$728$		0			0
$729$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$730$	0.390518	+	0.283728i	0.390518	+	0.283728i
$731$		0			0
$732$		0			0
$733$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	0.0627905	−	0.998027i	$-0.480000\pi$
$733$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	−0.929776	+	0.368125i	$0.880000\pi$
$734$	2.34039	−	1.70039i	2.34039	−	1.70039i
$735$		0			0
$736$	6.96299			6.96299
$737$	−1.06320	−	0.134314i	−1.06320	−	0.134314i
$738$		0			0
$739$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$739$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.541587	+	1.66683i	0.541587	+	1.66683i	0.728969	+	0.684547i	$0.240000\pi$
$743$	0.541587	+	1.66683i	0.541587	+	1.66683i	−0.187381	+	0.982287i	$0.560000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.618034			0.618034
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.728969	−	0.684547i	$-0.240000\pi$
$751$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.876307	−	0.481754i	$-0.160000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.108877	−	0.0791038i	−0.108877	−	0.0791038i
$756$		0			0
$757$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$757$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$758$		0			0
$759$		0			0
$760$	0.615377			0.615377
$761$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	0.535827	+	0.844328i	$0.320000\pi$
$761$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	−0.929776	+	0.368125i	$0.880000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−3.18523	−	2.31421i	−3.18523	−	2.31421i
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.637424	+	0.770513i	$0.720000\pi$
$773$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.929776	+	0.368125i	$0.880000\pi$
$774$		0			0
$775$	0.258996	+	0.797108i	0.258996	+	0.797108i
$776$	2.20877	+	6.79788i	2.20877	+	6.79788i
$777$		0			0
$778$	−1.36699	+	0.993173i	−1.36699	+	0.993173i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−3.79411	+	2.75658i	−3.79411	+	2.75658i
$785$		0			0
$786$		0			0
$787$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	−0.637424	−	0.770513i	$-0.720000\pi$
$787$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	0.0627905	−	0.998027i	$-0.480000\pi$
$788$		0			0
$789$		0			0
$790$	−0.0770013	+	0.236986i	−0.0770013	+	0.236986i
$791$		0			0
$792$	0.720253	+	3.77570i	0.720253	+	3.77570i
$793$		0			0
$794$	−0.378954	+	1.16630i	−0.378954	+	1.16630i
$795$		0			0
$796$		0			0
$797$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$797$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$798$		0			0
$799$		0			0
$800$	−4.34902	+	3.15975i	−4.34902	+	3.15975i
$801$	0.450527	−	1.38658i	0.450527	−	1.38658i
$802$		0			0
$803$	1.41213	+	1.32608i	1.41213	+	1.32608i
$804$		0			0
$805$		0			0
$806$	−2.39580	+	1.74065i	−2.39580	+	1.74065i
$807$		0			0
$808$	−1.73173	−	5.32971i	−1.73173	−	5.32971i
$809$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	0.876307	−	0.481754i	$-0.160000\pi$
$809$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	−0.992115	+	0.125333i	$0.960000\pi$
$810$	0.201592	+	0.146465i	0.201592	+	0.146465i
$811$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$811$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.0470631			−0.0470631
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.688925	+	0.500534i	0.688925	+	0.500534i	0.876307	−	0.481754i	$-0.160000\pi$
$821$	0.688925	+	0.500534i	0.688925	+	0.500534i	−0.187381	+	0.982287i	$0.560000\pi$
$822$		0			0
$823$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$823$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$827$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$828$	3.02932	−	2.20093i	3.02932	−	2.20093i
$829$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$829$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$830$	−0.0475894	−	0.146465i	−0.0475894	−	0.146465i
$831$		0			0
$832$	−8.71681	−	6.33313i	−8.71681	−	6.33313i
$833$		0			0
$834$		0			0
$835$	−0.106940			−0.106940
$836$	3.71491	+	0.469303i	3.71491	+	0.469303i
$837$		0			0
$838$		0			0
$839$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.535827	−	0.844328i	$-0.320000\pi$
$839$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.968583	−	0.248690i	$-0.0800000\pi$
$840$		0			0
$841$	0.309017	+	0.951057i	0.309017	+	0.951057i
$842$	1.14020	+	3.50919i	1.14020	+	3.50919i
$843$		0			0
$844$		0			0
$845$	0.0803940	−	0.247427i	0.0803940	−	0.247427i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$853$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$854$		0			0
$855$	0.129521	−	0.0941025i	0.129521	−	0.0941025i
$856$		0			0
$857$	−1.27485			−1.27485			−0.637424	−	0.770513i	$-0.720000\pi$
$857$	−1.27485			−1.27485			−0.637424	+	0.770513i	$0.720000\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	3.10969	+	2.25932i	3.10969	+	2.25932i
$863$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$863$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$864$		0			0
$865$		0			0
$866$	−2.04648	+	1.48686i	−2.04648	+	1.48686i
$867$		0			0
$868$		0			0
$869$	−0.425779	+	0.904827i	−0.425779	+	0.904827i
$870$		0			0
$871$	0.580394	−	1.78627i	0.580394	−	1.78627i
$872$		0			0
$873$	1.50441	+	1.09302i	1.50441	+	1.09302i
$874$	−0.996532	−	3.06701i	−0.996532	−	3.06701i
$875$		0			0
$876$		0			0
$877$	0.688925	−	0.500534i	0.688925	−	0.500534i	−0.187381	−	0.982287i	$-0.560000\pi$
$877$	0.688925	−	0.500534i	0.688925	−	0.500534i	0.876307	+	0.481754i	$0.160000\pi$
$878$	1.14020	−	3.50919i	1.14020	−	3.50919i
$879$		0			0
$880$	0.516099	−	0.283728i	0.516099	−	0.283728i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.613161	+	1.88711i	−0.613161	+	1.88711i
$883$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$883$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	0.535827	−	0.844328i	$-0.320000\pi$
$887$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	−0.637424	+	0.770513i	$0.720000\pi$
$888$		0			0
$889$		0			0
$890$	−0.363291			−0.363291
$891$	0.728969	+	0.684547i	0.728969	+	0.684547i
$892$	1.81527			1.81527
$893$		0			0
$894$		0			0
$895$	−0.0627905	−	0.0456200i	−0.0627905	−	0.0456200i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.893320	+	2.74936i	−0.893320	+	2.74936i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.178061	+	0.129369i	−0.178061	+	0.129369i
$906$		0			0
$907$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$907$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$908$		0			0
$909$	−1.17950	−	0.856954i	−1.17950	−	0.856954i
$910$		0			0
$911$	0.331159	−	1.01920i	0.331159	−	1.01920i	−0.637424	−	0.770513i	$-0.720000\pi$
$911$	0.331159	−	1.01920i	0.331159	−	1.01920i	0.968583	−	0.248690i	$-0.0800000\pi$
$912$		0			0
$913$	−0.115808	−	0.607087i	−0.115808	−	0.607087i
$914$	−2.89288			−2.89288
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	−0.992115	−	0.125333i	$-0.960000\pi$
$919$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	0.728969	−	0.684547i	$-0.240000\pi$
$920$	−0.497850	−	0.361710i	−0.497850	−	0.361710i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$929$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$930$		0			0
$931$	1.03137	+	0.749337i	1.03137	+	0.749337i
$932$		0			0
$933$		0			0
$934$	3.93717			3.93717
$935$		0			0
$936$	−6.73666			−6.73666
$937$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$937$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$941$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$	−2.74670	+	1.99559i	−2.74670	+	1.99559i
$950$	2.01421	+	1.46341i	2.01421	+	1.46341i
$951$		0			0
$952$		0			0
$953$	0.303189	+	0.220280i	0.303189	+	0.220280i	0.728969	−	0.684547i	$-0.240000\pi$
$953$	0.303189	+	0.220280i	0.303189	+	0.220280i	−0.425779	+	0.904827i	$0.640000\pi$
$954$		0			0
$955$		0			0
$956$	0.368852			0.368852
$957$		0			0
$958$	−1.22632			−1.22632
$959$		0			0
$960$		0			0
$961$	0.222357	+	0.161552i	0.222357	+	0.161552i
$962$		0			0
$963$		0			0
$964$	0.890518	+	0.646999i	0.890518	+	0.646999i
$965$		0			0
$966$		0			0
$967$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$967$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$968$	3.57386	−	1.41499i	3.57386	−	1.41499i
$969$		0			0
$970$	0.143188	−	0.440688i	0.143188	−	0.440688i
$971$	0.303189	−	0.220280i	0.303189	−	0.220280i	−0.425779	−	0.904827i	$-0.640000\pi$
$971$	0.303189	−	0.220280i	0.303189	−	0.220280i	0.728969	+	0.684547i	$0.240000\pi$
$972$		0			0
$973$		0			0
$974$	−1.07463	−	3.30738i	−1.07463	−	3.30738i
$975$		0			0
$976$		0			0
$977$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$977$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$978$		0			0
$979$	−1.44644	−	0.182728i	−1.44644	−	0.182728i
$980$	0.368852			0.368852
$981$		0			0
$982$		0			0
$983$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$983$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−2.02794	+	6.24136i	−2.02794	+	6.24136i
$989$		0			0
$990$	0.106096	−	0.225466i	0.106096	−	0.225466i
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	1.43726	−	4.42342i	1.43726	−	4.42342i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	0.0627905	−	0.998027i	$-0.480000\pi$
$997$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	−0.929776	+	0.368125i	$0.880000\pi$
$998$	−3.18523	+	2.31421i	−3.18523	+	2.31421i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	869.1.j.a.394.1	yes	20
11.5	even	5	inner	869.1.j.a.236.1	✓	20
79.78	odd	2	CM	869.1.j.a.394.1	yes	20
869.236	odd	10	inner	869.1.j.a.236.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
869.1.j.a.236.1	✓	20	11.5	even	5	inner
869.1.j.a.236.1	✓	20	869.236	odd	10	inner
869.1.j.a.394.1	yes	20	1.1	even	1	trivial
869.1.j.a.394.1	yes	20	79.78	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 869 = 11 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	869.j (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.613161 + 1.88711i) q^{2} +(-2.37622 - 1.72642i) q^{4} +(0.0388067 + 0.119435i) q^{5} +(3.10969 - 2.25932i) q^{8} +(0.309017 - 0.951057i) q^{9} +O(q^{10})\)	\(q+(-0.613161 + 1.88711i) q^{2} +(-2.37622 - 1.72642i) q^{4} +(0.0388067 + 0.119435i) q^{5} +(3.10969 - 2.25932i) q^{8} +(0.309017 - 0.951057i) q^{9} -0.249182 q^{10} +(-0.992115 - 0.125333i) q^{11} +(0.541587 - 1.66683i) q^{13} +(1.44922 + 4.46025i) q^{16} +(1.60528 + 1.16630i) q^{18} +(1.03137 - 0.749337i) q^{19} +(0.113982 - 0.350799i) q^{20} +(0.844844 - 1.79538i) q^{22} -1.27485 q^{23} +(0.796258 - 0.578516i) q^{25} +(2.81343 + 2.04407i) q^{26} +(-0.263146 + 0.809880i) q^{31} -5.46182 q^{32} +(-2.37622 + 1.72642i) q^{36} +(0.781687 + 2.40578i) q^{38} +(0.390518 + 0.283728i) q^{40} +(2.14110 + 2.01063i) q^{44} +0.125581 q^{45} +(0.781687 - 2.40578i) q^{46} +(0.309017 + 0.951057i) q^{49} +(0.603491 + 1.85735i) q^{50} +(-4.16459 + 3.02575i) q^{52} +(-0.0235315 - 0.123357i) q^{55} +(-1.36699 - 0.993173i) q^{62} +(1.89975 - 5.84683i) q^{64} +0.220095 q^{65} +1.07165 q^{67} +(-1.18779 - 3.65565i) q^{72} +(-1.56720 - 1.13864i) q^{73} -3.74444 q^{76} +(0.309017 - 0.951057i) q^{79} +(-0.476468 + 0.346175i) q^{80} +(-0.809017 - 0.587785i) q^{81} +(0.190983 + 0.587785i) q^{83} +(-3.36833 + 1.85176i) q^{88} +1.45794 q^{89} +(-0.0770013 + 0.236986i) q^{90} +(3.02932 + 2.20093i) q^{92} +(0.129521 + 0.0941025i) q^{95} +(-0.574633 + 1.76854i) q^{97} -1.98423 q^{98} +(-0.425779 + 0.904827i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 5 q^{4} - 5 q^{9}+O(q^{10}) \) 20 * q - 5 * q^4 - 5 * q^9	\( 20 q - 5 q^{4} - 5 q^{9} - 5 q^{16} + 15 q^{20} - 5 q^{22} - 5 q^{25} + 15 q^{26} - 10 q^{32} - 5 q^{36} - 10 q^{40} - 5 q^{49} - 10 q^{50} - 10 q^{62} - 5 q^{64} - 10 q^{76} - 5 q^{79} - 10 q^{80} - 5 q^{81} + 15 q^{83} - 5 q^{88} + 15 q^{92} + 15 q^{95}+O(q^{100}) \) 20 * q - 5 * q^4 - 5 * q^9 - 5 * q^16 + 15 * q^20 - 5 * q^22 - 5 * q^25 + 15 * q^26 - 10 * q^32 - 5 * q^36 - 10 * q^40 - 5 * q^49 - 10 * q^50 - 10 * q^62 - 5 * q^64 - 10 * q^76 - 5 * q^79 - 10 * q^80 - 5 * q^81 + 15 * q^83 - 5 * q^88 + 15 * q^92 + 15 * q^95

Introduction

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